2.  Calculus,10/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.

3.  Many real-world phenomena involve changing quantities-the speed of a rocket, the inflation of money (currency), the number of bacteria in a culture, the voltage of an electrical signal, etc.  A “derivative” is the mathematical tool for studying the rate at which one value changes relative to another.  Tangent lines relate this change to slope.

4.  As points P and Q on this graph get closer and closer together, the slope of the secant line through P and Q gets closer to the slope of the tangent line which is only at P.  From Geometry, a secant line crosses twice and a tangent line touches once.

5.  The slope of the secant line comes from Algebra I, m = rise/run =  This book uses different notation, but it means the same thing.

6.  The slope of the tangent line is only through one point so we cannot use the same equation. Instead, we must calculate the limit as point Q approaches point P.

7.  One of the important themes in calculus is the study of motion. To describe motion we discuss speed and direction of travel which, together, comprise velocity.  For this section, we will only consider motion along a line (rectilinear motion).

9.  Instead of an average, we often want to determine velocity at a specific instant in time. It is like finding the slope of the tangent line vs. finding the slope of the secant line.  We only have one time, so we cannot subtract. Instead, we must find the limit as we get closer and closer to the exact time we are looking for.

10.  Derivatives are the primary mathematical tool that we use to study and calculate rates of change in a variety of different applications and can be used to calculate instantaneous rates of change.

11.  There are several different formulas you can use which all look different and mean the same thing. Pick the one that you like best and stick with it.  This is the one we used last time: › Derivative =  Another option:

13.  You got the same result using any method for the derivative which is the slope of the tangent line for any value of x.  We can use it to find the slope at any given value of x:

14.  This looks intimidating, but it just means to use point slope form once you have the slope.

16.  It is possible that the limit that defines the derivative of a function may not exist at certain points in the domain of the function. When that happens, the derivative is undefined.

17.  Two common ways in which a function that is continuous can fail to be differentiable at certain values of x are when the following exist: › Corner points › Points of vertical tangency

18.  As you can see in the picture, the one sided limits of the slope of the secant line from the two sides of x 0 are not equal so the two sided limit does not exist.

19.  As you can see in the picture, the limits of the slope of the secant line approach +/- infinity which is not a measurable slope.

20.  The previous two slides show that functions can be continuous at certain points where they are not differentiable (meaning you cannot take the derivative there).  It is also true that:  The proof is interesting, read about it and some related history on page 149.

21.  We will talk more about these, but you should know that these mean the same thing:  Also, there are other formulas that some people use to find the derivative:  I tend to like the last one, but make your own choice.

22.  If you consider the graph of y=2 (for example), or y = any constant, it is a horizontal line whose slope is zero (see graph page 155 if you have a question).  Since the derivative of y=2 equals its slope, the derivative must be 0.  y’ = f’(x) = 0

23.  We have taken the derivative of linear (x 1 ), quadratic (x 2 ), and cubic (x 3 ) functions in previous sections and talked about the “shortcut rule” or power rule.  There is a more formal proof in the middle of page 156.

24.  We can use a proof very similar to the one on the previous slide (see page 158) to find the derivatives of sums and differences.  Which just means that the derivative of a sum/difference equals the sum/difference of the derivatives (as it did with limits).

25.  The second derivative is the derivative of the first derivative and all of the same rules apply.  As long as we have differentiability, we can continue the process of differentiating to obtain third, fourth, fifth, and higher derivatives of a given function.

26.  We will learn a method for differentiating (taking the derivative of) products.  NOTE: The derivative of the product is NOT equal to the product of the derivatives. An example on the next page will show why.

27.  If we say that f(x) = x and g(x) = x 2 and we take their derivatives, we get that f’(x) = 1 and g’(x) = 2x by using the power rule on each function.  We would like to think that › [f(x) * g(x)]’ = f’(x) * g’(x)] WRONG › = 1 * 2x = 2x WRONG  It does not because f(x) * g(x) = x* x 2 = x 3 which has a derivative of f’(x) = 3x 2, NOT 2x.

30.  Since addition is “commutative”, it does not matter what order you add in and 3+7 = 7+3.  When we get to the quotient rule, it has subtraction in it and subtraction is not commutative.  Therefore, I highly suggest you use the product rule in this form: › (f*g)’ = f’*g +g’ *f

31.  Everything I wrote about where the product rule comes from and why also applies to the quotient rule.

32.  The proof is on pages 165-166.  The notation I like to use is:  (f/g)’ = f’g - g’f g 2

34.  You can prove the derivatives of cos x and tan x similarly to the proof for sin x on the following slide.

38.  The “Chain Rule” is the method we use when taking the derivative of a composition of functions f(g(x)).  y’ = dy/dx is obtained by a two-link “chain” of simpler derivatives

39.  We will use an alternative version of the chain rule on a later slide to find the same derivative. Decide which method you prefer.

40.  can be frustrating during longer problems due to too many variables so you may want to use this instead.  They mean the same thing:

42.  Make sure you only do one step at a time.

44.  If you go to YouTube and search Cruzers baton 1987, you should be able to find my team’s video from the world championships in Paris if you are interested. I am the really tall and white person towards the right. It looks like this: